A290372 10-adic integer x = ...5807 satisfying x^5 = x.

7, 0, 8, 5, 9, 2, 6, 6, 6, 1, 8, 5, 3, 0, 0, 7, 4, 8, 1, 1, 4, 2, 6, 8, 7, 8, 7, 3, 2, 4, 1, 6, 1, 5, 1, 1, 5, 4, 5, 0, 2, 2, 9, 0, 6, 9, 2, 1, 7, 4, 7, 2, 2, 2, 2, 1, 7, 5, 8, 7, 8, 5, 2, 4, 8, 0, 6, 9, 6, 4, 4, 8, 5, 8, 3, 0, 8, 6, 5, 2, 5, 0, 6, 6, 9, 9, 1, 5

Offset: 0

Author

Seiichi Manyama, Jul 28 2017

Keywords

• nonn
• base

Comments

Also x^2 = A091661.

Examples

			     7^5 -    7 == 0 mod 10,
     7^5 -    7 == 0 mod 10^2,
   807^5 -  807 == 0 mod 10^3,
  5807^5 - 5807 == 0 mod 10^4.
From _Seiichi Manyama_, Aug 01 2019: (Start)
  2^(5^0) - 5^(2^0) ==    7 mod 10,
  2^(5^1) - 5^(2^1) ==    7 mod 10^2,
  2^(5^2) - 5^(2^2) ==  807 mod 10^3,
  2^(5^3) - 5^(2^3) == 5807 mod 10^4. (End)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..9999

Crossrefs

Cf. A120817, A120818, A290373, A290374, A290375.

Cf. A091661, A018247.

Programs

Ruby

def P(n)
  s1, s2 = 2, 8
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k1, k2 = j * m + s1, (9 - j) * m + s2
      if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
        s1, s2 = k1, k2
        break
      end
    }
  }
  s1
end
def Q(s, n)
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k = j * m + s
      if (k ** 2 - k) % (m * 10) == 0
        s = k
        break
      end
    }
  }
  s
end
def A290372(n)
  str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A290372(100)

Formula

p = A120817 = ...186432, q = A018247 = ...890625, x = p - q = ...295807.